| L(s) = 1 | − 6·5-s − 4·7-s + 2·13-s − 12·19-s + 17·25-s + 6·29-s + 12·31-s + 24·35-s + 16·37-s + 6·41-s − 4·43-s − 8·47-s + 4·49-s − 2·53-s + 4·59-s + 2·61-s − 12·65-s − 8·67-s + 4·71-s + 6·73-s − 24·79-s − 9·81-s − 4·83-s − 2·89-s − 8·91-s + 72·95-s + 6·97-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 1.51·7-s + 0.554·13-s − 2.75·19-s + 17/5·25-s + 1.11·29-s + 2.15·31-s + 4.05·35-s + 2.63·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 4/7·49-s − 0.274·53-s + 0.520·59-s + 0.256·61-s − 1.48·65-s − 0.977·67-s + 0.474·71-s + 0.702·73-s − 2.70·79-s − 81-s − 0.439·83-s − 0.211·89-s − 0.838·91-s + 7.38·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7792931091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7792931091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 23 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 12 T + 68 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 67 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 104 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 140 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 146 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 179 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.85111964910540416398113020283, −7.78098523552231746436921124402, −7.35419630103221897974820391213, −6.83541994547435757093443458090, −6.63609082231443004424623561148, −6.29680431745563089588601182998, −6.05573371932215340017507843255, −5.83657206822917314483643711219, −4.69976944413607947119600354203, −4.68850246772116126878247259151, −4.46131312179153461526264297037, −3.99435764542081079557076025353, −3.80485494160127383364013876299, −3.35848250925148638683073740842, −2.86650522883089233885702366067, −2.75243464786618541414837244980, −2.15862349297678166053309177281, −1.29578320510573281359593748324, −0.57114632521050694568744861385, −0.39090756518103539303874738907,
0.39090756518103539303874738907, 0.57114632521050694568744861385, 1.29578320510573281359593748324, 2.15862349297678166053309177281, 2.75243464786618541414837244980, 2.86650522883089233885702366067, 3.35848250925148638683073740842, 3.80485494160127383364013876299, 3.99435764542081079557076025353, 4.46131312179153461526264297037, 4.68850246772116126878247259151, 4.69976944413607947119600354203, 5.83657206822917314483643711219, 6.05573371932215340017507843255, 6.29680431745563089588601182998, 6.63609082231443004424623561148, 6.83541994547435757093443458090, 7.35419630103221897974820391213, 7.78098523552231746436921124402, 7.85111964910540416398113020283
